Let f be a twice differentiable function on R with f” continuous on [0, 1] such that
the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4).
Prove that there exists an xo in (0, 1) such that f”(xo) = 0.
Here is what I have so far....
the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4)
f(c1) (1-0) = 2 f(c2) (3/4 - 1/4)
for c1 belonging to (0, 1) and c2 belonging to (1/4, 3/4)
f(c1) = f(c2)
By Rolle's Theorem, there exists c3 that belongs to (c1, c2) such that f ' (c3)=0
I need another part similar to this so that the derivitive of these which would be f " (xo) = 0.
Any suggestions are welcome.